Sharp empirical Bernstein bounds for the variance of bounded random variables

TL;DR

This paper introduces new empirical Bernstein inequalities for bounded random variables' variance, applicable in both batch and sequential settings, with asymptotic sharpness and extensions to Hilbert spaces.

Contribution

The paper presents novel empirical Bernstein bounds that are valid under constant conditional variance and mean without independence assumptions, improving upon existing inequalities.

Findings

Bounds are asymptotically sharp for iid data.

The inequalities outperform traditional self-bounding variance bounds.

Extensions to Hilbert spaces demonstrate broad applicability.

Abstract

We develop novel empirical Bernstein inequalities for the variance of bounded random variables. Our inequalities hold under constant conditional variance and mean, without further assumptions like independence or identical distribution of the random variables, making them suitable for sequential decision making contexts. The results are instantiated for both the batch setting (where the sample size is fixed) and the sequential setting (where the sample size is a stopping time). Our bounds are asymptotically sharp: when the data are iid, our CI adpats optimally to both unknown mean $\mu$ and unknown $\mathbb{V}[(X-\mu)^2]$, meaning that the first order term of our CI exactly matches that of the oracle Bernstein inequality which knows those quantities. We compare our results to a widely used (non-sharp) concentration inequality for the variance based on self-bounding random variables, showing both the theoretical gains and improved empirical performance of our approach. We finally extend our methods to work in any separable Hilbert space.